Let $\mathrm{Aut}(\Gamma)$ be the automorphism group of a graph $\Gamma$. Also suppose that $\mathrm{Cay}(G)$ is the Cayley graph of a group $G$. Consider the following chain:

$\Gamma \to \mathrm{Aut}(\Gamma) \to \mathrm{Cay}(\mathrm{Aut}(\Gamma)) \to \mathrm{Aut}(\mathrm{Cay}(\mathrm{Aut}(\Gamma))) \to \mathrm{Cay}(\mathrm{Aut}(\mathrm{Cay}(\mathrm{Aut}(\Gamma)))) \to \ldots$ ,

where arrow $A\to B$ just means making $B$ from $A$. For example $\Gamma \to \mathrm{Aut}(\Gamma)$ means that the automorphism group $\mathrm{Aut}(\Gamma)$ is constructed based on $\Gamma$.
Is there any graph $\Gamma$ for which the above chains stops, i.e. after $i$ steps

$\Gamma \cong \mathrm{Cay}(\mathrm{Aut}(\mathrm{Cay}(\cdots(\mathrm{Aut}(\Gamma))))$, 

as two graphs (the notation $\cong$ means graph isomorphism)?

What about the case that starts with a group $G$, i.e. is there any group $G$ such that

$G \cong \mathrm{Aut}(\mathrm{Cay}(\mathrm{Aut}(\cdots(\mathrm{Cay}(G))))$,

as two groups (the notation $\cong$ means group isomorphism and $\mathrm{Aut}(G)$ is the automorphism group of the group $G$ here)?